Why are the neutron and proton masses nearly equal?

Question

The neutron to proton mass ratio is nearly one. Is there some fundamental reason from this or this simply a coincidence?

Answer 1

If you calculate the ratio between the proton mass and its constituent quarks, you'll see that the quarks actually account for only 1.0% of the proton mass. A similar calculation for a neutron shows that quark masses account for 1.3% of the neutron mass. Thus for both of these particles, 99% of the mass is not simply the sum of masses of the subatomic particles.

So where does the mass of these particles come from then? It can be interpreted as the interaction energy of the subatomic particles (think $E=mc^2$ backwards); and since protons and neutrons both consist of a relativistic quark-gluon mixture which interacts through the strong force in nearly the same way, the interaction energies of the quarks in a proton and neutron are very close.

PS: For another perspective on the importance of the interaction energies of the quarks: if you have heard of the particles $\Delta^+$ and $\Delta^0$ before (see Delta baryon), these particles have the quark contents $uud$ and $udd$ respectively, so in other words exactly the same quark content as protons and neutrons. The only difference is that these particles are 30% heavier than the more common protons and neutrons. You can therefore think of the $\Delta^+$ as an excited state of a proton, and the $\Delta^0$ as an excited neutron, where the increase in energy associated with this excitation ends up increasing the mass of the particle by 30%!

Answer 2

Both protons and neutrons are made up of two types of quarks: up (u) and down (d). Protons are uud and neutrons udd. QCD, the strong force binds these quarks together into protons and neutrons (technically, the binding involves a "sea" of gluons and quark-antiquark pairs). There is an approximate symmetry of QCD called isospin. Both the u and d quarks are very light compared to the intrinsic scale of QCD (an energy scale called $\Lambda_{QCD} \sim 100 MeV$. In the approximation that the u and d quark masses are the same*, which isn't bad, QCD treats the u and d quarks identically. In this symmetry, protons and neutrons are equivalent.

• There might be some technicality about the s quark mass in the actual definition.